We introduce a function on sequences, which we call the Pfaffian
transform, using the Pfaffian of a skew-symmetric matrix. We establish
several basic properties of the Pfaffian transform, and we use the
transfer matrix method to show that the set of sequences with rational
generating functions is closed under the Pfaffian transform. We
conclude by computing the Pfaffian transform of a variety of sequences,
including geometric sequences, the sequence of Fibonacci numbers, the
sequence of Pell numbers, the sequence of Jacobsthal numbers, and the
sequence of Tribonacci numbers. Throughout we describe a
generalization of our results to Pfaffians of skew-symmetric matrices
whose entries satisfy a Pascal-like relation.